On enhanced sensing of chiral molecules in optical cavitiesPhilip Scott,1, a) Xavier Garcia-Santiago,2, 3 Dominik Beutel,2 Carsten Rockstuhl,2, 4 Martin Wegener,1, 4 and IvanFernandez-Corbaton4, b)1)Institute of Applied Physics, Karlsruhe Institute of Technology, 76128 Karlsruhe, Germany2)Institute of Theoretical Solid State Physics, Karlsruhe Institute of Technology, 76128 Karlsruhe,Germany3)JCMWave GmbH, 14050 Berlin, Germany4)Institute of Nanotechnology, Karlsruhe Institute of Technology, 76021 Karlsruhe, Germany

(Dated: 25 July 2021)

The differential response of chiral molecules to incident left- and right- handed circularly polarized light is used forsensing the handedness of molecules. Currently, significant effort is directed towards enhancing weak differentialsignals from the molecules, with the goal of extending the capabilities of chiral spectrometers to lower molecularconcentrations or small analyte volumes. Previously, optical cavities for enhancing vibrational circular dichroismhave been introduced. Their enhancements are mediated by helicity-preserving cavity modes which maintain thehandedness of light due to their degenerate TE and TM components. In this article, we simplify the design of thecavity, and numerically compare it with the previous one using an improved model for the response of chiral molecules.We use parameters of molecular resonances to show that the cavities are capable of bringing the vibrational circulardichroism signal over the detection threshold of typical spectrometers for concentrations that are one to three orders ofmagnitude smaller than those needed without the cavities, for a fixed analyte volume. Frequency resolutions of currentspectrometers result in enhancements of more than one order (two orders) of magnitude for the new (previous) design.With improved frequency resolution, the new design achieves enhancements of three orders of magnitude. We showthat the TE/TM degeneracy in perfectly helicity preserving modes is lifted by factors that are inherent to the cavities.More surprisingly, this degeneracy is also lifted by the molecules themselves due to their lack of electromagnetic dualitysymmetry, that is, due to the partial change of helicity during the light-molecule interactions.

Keywords: Chirality, sensing, helicity preserving scattering, optical cavities

I. INTRODUCTION AND SUMMARY

The fundamental biomolecules in living organisms arechiral. Therefore, the effect that the two enantiomers of achiral molecule have on them can be very different1. Sensingthe dominant handedness of chiral molecules is thus animportant task in chemistry, biology, and pharmacology. Twoof the phenomena that are exploited for this task are thedifferential absorption of circularly polarized light, knownas circular dichroism (CD), and the polarization rotation oflinearly polarized light, known as optical rotation (OR). TheCD, like the OR, is equal in magnitude but opposite in signfor similar ensembles of opposite enantiomers2. The chiralityof the molecules manifests itself when electromagneticradiation induces electronic molecular transitions, molecularvibrations, or molecular rotations. The three differentmechanisms are activated by illumination frequencies in thenear-UV, infrared, and the GHz region, respectively3. In allthree cases, the inherent weak chiral response of the moleculeslimits the usability of existing spectrometers for measuringthe CD signal as the molecular concentration and/or theanalyte volume decrease. Enhancing the differential signalis then necessary. The photonics community has taken on thischallenge with a recent stream of theoretical and experimentalstudies4–39.

a)Electronic mail: philip.scott@kit.edub)Electronic mail: ivan.fernandez-corbaton@kit.edu

Enhanced sensitivity for the measurement of opticalrotation has been experimentally shown in Cavity Ring DownPolarimetry (CRDP) setups4,5,8,14,15, where light pulses passthrough the sample multiple times, and quarter-wave platesand/or Faraday rotators are used to overcome previouslyencountered difficulties40. In such setups, the single-passoptical pathlength is on the order of one meter. At muchsmaller scales, one of the main strategies is to increasethe light-molecule interaction by means of photonic micro-structures that resonantly enhance the fields. In this context,the electromagnetic helicity24,41–64 has proven to be a usefulquantity in the analysis and design of systems for enhancedsensing of chiral molecules29,30. The electromagnetic helicitycan be seen as the generalization of the concept of circularpolarization to general electromagnetic fields, including nearfields, evanescent fields, and cavity modes. Achiral andresonant systems that do not change the helicity of theincident light can be seen as the resonant version of typicalchiro-optical spectrometers29. Achirality is needed to avoidsignal distortions including non-zero signals from achiralanalytes13,17,25,29. While helicity preserving resonances arenot a must, they are optimal under some conditions29 forenhancing the CD signal. Several achiral systems featuringhelicity preserving resonances for enhanced sensing of chiralmolecules have been reported22,29,30,33,35,37,39. Helicitypreservation is achieved if and only if the TE and TMresponses of the system are identical (see Chap. 2.4 inRef. 65). For resonant systems, this implies the degeneracyof TE and TM modes. Very recently, we have reportedthe theoretical design of an optical cavity featuring helicity

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FIG. 1: (a-c) Sketches of systems for CD enhancement. (a) A single-array of silicon disks connected by rods, without asubstrate. (b) A cavity formed by two silicon disk arrays arranged in hexagonal lattices where r =1.92 µm, h =1.32 µm, anda =5.76 µm, with substrate thickness ds =2 µm, and cavity length L. (c) A cavity formed by an array of silicon disks with thesame parameters as in (b), and a homogeneous slab of silicon with thickness d =1 µm. (d) Ray diagram showing the path ofthe light as it passes through the double-array cavity depicted in (b). The behavior on the single-array cavity depicted in (c) isessentially the same. The light impinges from above in this depiction (red). The zeroth diffraction orders of the two arrays exciteFabry-Perot modes in the cavity (black), which are not useful for CD enhancement. The CD enhancing modes are due to thefirst diffraction orders, which produce modes of large transverse momentum (green). In these modes, the light inside the cavitybounces without changing handedness at grazing angles between the two disk arrays until it is diffracted out of the cavity by oneof the arrays (blue). The two arrays are assumed to be periodic in the lateral directions.

preserving modes for enhancing the vibrational circulardichroism (VCD) signal of chiral molecules30. The cavity,sketched in Fig. 1(b), consists of two parallel mirrors madefrom arrays of high permittivity dielectric disks. The disksare placed on top of substrates and are arranged in a periodichexagonal pattern. The two mirrors are placed parallel toeach other at distances on the order of tens of micrometers,thereby forming a cavity that is filled by the solution of

chiral molecules. In contrast to many other designs wherethe CD enhancements occur only in close proximity to somestructured photonic material, the differential absorption inour proposed cavity is enhanced practically across the wholevolume of the analyte inside the cavity. The modal fieldsof almost pure helicity corresponding to the first diffractionorders of the cavity provide average enhancements of theVCD signal of over two orders of magnitude for cavity lengths

3

of a few tens of micrometers. This, together with the intrinsicspectral tunability of a cavity make the design in Ref. 30very appealing for experimental realization. Figure 2 depictsa possible experimental setup. At this point, the refinementof the theoretical models becomes necessary for obtainingprecise performance predictions, and the consideration ofalternative designs of increased operational simplicity iswarranted.

In this article, we compare two optical cavity designsfor CD enhancement: Our previous design featuring twoidentical arrays of tailored silicon disks, which we will call“double-array cavity”, and a new simplified design whereone of the arrays is substituted by a thin, homogeneoussilicon slab, which we will call “single-array cavity”. Thisseemingly minor modification constitutes a very valuablesimplification experimentally because it eliminates the needfor lateral alignment of the two silicon-disk arrays on a deepsub-micrometer scale. The performance of the cavities isaccurately predicted by using a realistic model for the opticalresponse of the solution of chiral molecules. The parametersof the model can be fixed using existing spectroscopicmeasurements of the targeted resonance for the desired chiralmolecule. While we will here focus on molecular solutionsin the liquid phase, the same modeling strategy can beused for the gas phase. In particular, the model accountsfor the duality symmetry-breaking of the molecules50,66,that is, for the partial change of electromagnetic helicityduring light-molecule interaction. We show that the degreeof duality breaking increases as the Kuhn’s dissymmetryfactor67 (g) for the resonance decreases. This symmetrybreaking, which is completely inconsequential in typicalCD spectrometers, can affect the modes of the cavity dueto the long light-matter interaction times. We show thatthe ideal TE/TM degeneracy in perfectly helicity preservingmodes is disturbed by factors that are inherent to the designof the cavities, and also by the lack of electromagneticduality symmetry of the chiral molecules. These disturbancescause a splitting of the TE and TM modes. As thesplitting grows, the enhancement decreases and the spectralsignatures of the CD signals progressively change from asingle helicity preserving resonance to two sharper helicitypreserving features separated by a helicity flipping region,in which the sign of the signal changes. The contributionof the molecules to the splitting increases as the molecularconcentration divided by the Kuhn’s dissymmetry factor ofthe molecular resonance increases, and as the linewidth of thespectral features of the cavity response decreases. With theseeffects taken into account, we show that both cavity designscan provide VCD enhancements of more than two orders ofmagnitude, which we exemplify for a particular resonance ofthe binol molecule. For the double-array cavity, the standardfrequency resolution of current spectrometers is much finerthan the width of the enhancement lines. For the single-array cavity, the linewidth of the helicity-preserving featuresthat achieve enhancements of three orders of magnitudeare comparable to the standard frequency resolution ofcommercial VCD spectrometers. The single-array cavity canalso be operated exploiting much wider features that achieve

more than one order of magnitude of negative enhancement,where the sign of the CD signal is deterministically changed.For the simplest operation of the cavities when targetinga particular resonance, the concentration should be lowenough to avoid the more severe effects of the splitting andlarge enough so that the CD signal is above the sensitivitythreshold of the measurement apparatus. The upper limit inthe concentration could be avoided by the use of parameterestimation techniques exploiting that the effects of the TE/TMsplittings in the CD signal can be accurately reproducedusing a simple analytical model. Our results imply thatthe effects of the duality breaking of the molecules onthe response of helicity-preserving optical cavities are lesspronounced for electronic CD and more pronounced forrotational CD, with vibrational CD lying in the middle.This statement follows from the connection between dualitysymmetry breaking and Kuhn’s dissymmetry factor, and theknown scaling of g for each kind of CD: The largest valuesof g in electronic, vibrational, and rotational CD spectroscopyare of the order 10−1,10−3, and 10−5, respectively68–70, andthe typical values are at least an order of magnitude lower ineach case. The rest of the article is organized as follows.

In Sec. II, we make the idealized assumption that themolecules do not change the helicity of the light that theyinteract with and study the undisturbed response of the twocavities. We identify the different causes of TE/TM splittingthat degrade the helicity preserving properties of the cavitymodes and change the spectral shapes of the enhancementlines. We show that these changes can be modeled bythe composition of a TE and a TM resonance with varyingfrequency detuning. In Sec. III we develop a model for theconstitutive relations of the solution of a chiral molecule forthe frequency region containing a molecular resonance. InSec. IV we use the model to target a particular resonanceof binol in its VCD spectrum, and to study the effects ofthe duality breaking of the molecules on the two cavities.Section V contains the concluding remarks.

II. CAVITY DESIGN COMPARISON

We consider the three systems depicted in Fig. 1. Asingle planar hexagonal array of silicon disks connected byrods (without a substrate), a cavity formed by two parallelhexagonal arrays of silicon disks, and a cavity formed by anarray of silicon disks and a homogeneous silicon slab. Thenot-to-scale molecule drawings in Fig. 1 represent the solutionof chiral molecules. The disk arrays and the silicon slabdefining the cavities are placed on substrates, and the mediumoutside the two cavities is air. The systems are sequentiallyilluminated from the top by perpendicularly incident planewaves of opposite polarization handedness.

The geometrical parameters of the single planar arrayin Fig. 1(a) can be optimized to produce a resonant andhelicity preserving response upon illumination at normalincidence29,33. The lattice pitch a

√4/3 is set to ensure

that only the zeroth diffraction order is allowed to propagatethrough the solution. At the resonance frequency, the incident

4

FIG. 2: An artistic scheme of the components of a CD spectrometer setup with our CD enhancing cavity surrounding thesample. An enlarged version of this later part of the setup can be seen in the inset, where the blue disks indicate the substrateswith the metasurface mirrors on top of them. MSC-1: Metasurface mirror 1, MSC-2: Metasurface mirror 2, L: Light source, SM:Spectrometer, P: Linear polarizer, PEM: Photoelastic modulator, S: Sample encased in our CD enhancing cavity, D: Detector.The light emerges from the light source, enters a spectrometer, and then travels through a linear polarizer. Next, the photoelasticmodulator continuously flips the light between left- and right-handed circularly polarized-light at a certain modulation frequency.The light interacts with the sample and finally hits the detector, which sends the information to a computer for analysis. The redarrows mark several points in the light path for which the polarization of light is depicted with black arrows.

circularly polarized light produces strong near fields of purehelicity attached to the disks. These near-fields result in CDenhancement factors that reach ≈ 20 at the surface of thedisks, but then drop quickly with increasing distance from thedisks. Therefore, only the molecules in the immediate vicinityof the disks experience a significant effect. While evaporationof chiral molecules onto the silicon disks34 provides a meansto exploit the near-field effect, the practical applicability ofsingle-array sensors will be limited by the fact that the CDsignal would not be enhanced over macroscopic volumesof analyte. This problem is solved by the double-arraycavity in Fig. 1(b), where the helicity preserving resonancesare achieved in a quite different way30. As illustrated inFig. 1(d), the arrays of silicon disks and their lattice pitchare designed such that circularly polarized light impingingonto the cavity under normal incidence is diffracted into afirst order towards large angles approaching 90 degrees whilepreserving the helicity of the light. Next, when the cavityresonance condition is met for the diffracted light, the lightinside the cavity bounces obliquely between the two diskarrays at grazing angles, leading to a large interaction timewith the chiral molecules. Crucially, the helicity is preservedbecause the TE- and TM-polarizations of light become nearlydegenerate in their behavior for grazing incidence onto the

disk arrays. Finally, the light is diffracted out of the cavityin a normal direction by the silicon disk arrays and can bedetected. Importantly, the strong modal fields of almost purehelicity inside the cavity are not attached to the disks, but arespread across the entire volume between the two silicon diskarrays. Resonant enhancement factors exceeding a factor of2000 at some points inside the cavity can be reached. Whenaveraging the enhancement factors across the volume of theentire cavity, enhancement factors exceeding 100 at cavitylengths of ≈ 20µm have been numerically demonstrated inRef. 30 for idealized molecules with a large degree of helicitypreservation. For the experimental realization of the cavity,the need for precise lateral alignment of the two silicondisk arrays must be emphasized: Misalignments result in anoverall chiral system which distorts and can overwhelm thedesired signal from the molecules (see Ref. 30, App. IV).In practice, such alignment can pose a substantial challenge.This difficulty motivates an alternative new design for thecavity, depicted in Fig. 1 (c), where one of the two arrays isreplaced by an thin, homogeneous silicon film that acts as aregular mirror under conditions of grazing incidence. The ideabehind the simplification is that only one silicon disk arrayis needed for coupling of the internal cavity modes featuringgrazing angles to the external perpendicular directions of

5

FIG. 3: Comparison plots of the double-array cavity and the single-array cavity. Panels (a) and (c) show the TCD and ACDenhancements of the double-array cavity, respectively. Panels (b) and (d) show the TCD and ACD enhancements of the siliconslab and array cavity, respectively. The false color scale has been truncated from below at the level of 10−2.5 in linear units, andthere exist negative enhancement values in some of the dark blue regions. The red crosses mark the data cuts shown in Fig. 5.

illumination and measurement. Since the new design hasonly a single disk array, it does not need lateral alignment.Both designs still need the two sides of the cavity to be asparallel as possible to each other. In our simulations, we haveassumed that the two sides are perfectly parallel. In practice,this can be controlled by using a set of three linear actuatorsin a triangular formation at the edges of the cavity to adjustone of the sides (see the inset in Fig. 2). In our simulations,we also assume that the arrays and the silicon slab extend toinfinity in the lateral directions. In practice, the lateral extentshould be large enough so that the edge effects are negligible.

We now explore the simplified single-array cavity designand compare it to the double-array cavity. We use thesame T-matrix-based simulation of laterally infinitely periodicarrays71 that we used in Ref. 30 for numerical evaluations.The T-matrices of the silicon disks are obtained with thehelp of JCMsuite72 assuming that the relative permittivityof silicon is 11.9. We also assume that the substrates andthe solvent have a relative permittivity equal to 2.14. Thesolution of chiral molecules is modeled using the Condon-Tellegen constitutive relations for monochromatic fields with

an implicit exp(−iωt) time dependence

Dω(r) = εωeffE

ω(r)+ iκω

effc0

Hω(r),

Bω(r) = µωeffH

ω(r)− iκω

effc0

Eω(r),(1)

where SI units are used and we assume for now a non-dispersive response of the molecules: εω

eff = 2.14(1 +

i10−4)ε0, µωeff = (1+ i10−4)µ0, and κω

eff = i10−4, where ε0(µ0) is the vacuum permittivity (permeability). We notethat the assumed values meet the equality Z2

solεωeff = µω

eff,where Zsol is the optical impedance of the solvent. Thisimplies that the molecules have duality symmetry66, that is,that they do not change the helicity of the light that theyinteract with. We will later analyze the effect of removingthis idealization. Figure 3 shows the absorption CD (ACD)and transmission CD (TCD) enhancements for both cavitiesas a function of the frequency f in units of THz and cavitylength L in µm. The definition of ACD is ACD = P+−P−

2P0,

where P+ and P− are the outgoing powers for the subsequentleft-handed (“+”) and right-handed (“-”) circularly polarized

6

FIG. 4: Panels (a) and (c) show the TCD enhancement of the single- and double-array cavity, respectively, as a function offrequency for a cavity length L=21.52 µm. The TCD enhancement is shown here in linear units. In each case, the insets showthe lowest frequency mode in more detail. Panels (b) and (d) show the normalized helicity content resulting from a model wherea TE and a TM resonance with linewidths equal to γc are located ∆ f Hz apart (see text).

incident light, assuming identical incident powers equal toP0. The outgoing power is the sum of the powers of thetransmitted and reflected fields. We recall that the array ofdisks is diffracting in the solvent but not in air, so only theforward and backward directions perpendicular to the planeof the arrays can send energy outside of the cavity (P± =Pfwd± +Pbwd

± ). The definition of TCD is

TCD =Pfwd+ −Pfwd

−Pfwd+ +Pfwd

−, (2)

where only the transmitted powers Pfwd± need to be detected.

The ACD (TCD) enhancement factor is defined by theACD (TCD) signal with the enhancing cavity divided bythat without the enhancing cavity. We note that the falsecolor scale of Fig. 3 has been truncated from below at thelevel of 10−2.5 in linear units, and that there exist negativeenhancement values in some of the dark blue regions, whichwe analyze later.

Figure 3 shows very similar positions of the high-

enhancement lines for the two cavities. As explained inRef. 30, the lines in the (L, f ) parameter space correspondto different cavity modes excited by the first diffraction ordersof the hexagonal lattice upon normal incidence. The positionof the enhancement lines depends on several parameters (seeEq. (B8) in Ref. 30): The cavity length L, the mode number,the lattice spacing a, and the refractive indices of the disksand the solvent. These parameters can be modified to tunethe frequency response of the cavities and enhance the CD ofa given chiral molecule resonance. While the position of thelines is essentially the same in both cavities, their structureis different. One difference is that the enhancement along agiven line is more uniform for the single-array cavity. Theenhancement variations along the double-array cavity linesare due to the coupling between the first diffraction ordersand the zeroth diffraction orders30 (Fabry-Perot modes). Thelatter modes feature helicity flipping perpendicular reflectionsbetween the cavity walls (see Fig. 1(d)). When the (L, f )modal condition lines of those Fabry-Perot modes cross a firstdiffraction order line, a mixed mode forms if the differential

7

phase that the two modes accumulate in a half round-trip isan integer multiple of 2π30,73. The mixed modes loose thehelicity preserving properties of the grazing first order modes,which causes a sharp degradation of the helicity purity ofthe field, severely disrupting the enhancement. In the single-array cavity, the two half round-trips of a given mode arenot equivalent and, as a consequence, the condition for thecreation of a mixed mode is not fulfilled. Ultimately, this isdue to the lack of parity symmetry (r→ −r) of the single-array cavity. The substitution of one of the disk arrays withthe silicon slab breaks the parity symmetry of the double-array cavity. One mirror symmetry plane is also lost. Thissymmetry breaking is a consequential difference between thetwo cavity designs.

The presence of reduced incidence angles due to the siliconslab is another consequential difference. Due to the differencebetween the refractive indices of the solvent and the siliconfilm, the propagation of light inside the silicon slab occurs atangles that are much smaller than in the solution. Then, thereflection off the silicon-substrate interface near the bottom ofFig. 1(c) is never near-grazing anymore for the first diffractionorders. For example, when the angle of incidence onto thesolution-silicon interface tends to 90 degrees, the angle ofincidence onto the silicon-substrate interface tends to 25.1degrees. The small angles of incidence cause the splittingof the TE- and TM- polarizations, degrading the helicitypreservation of the reflections and hence of the modes. Thissplitting is behind the other salient difference in Fig. 3: Eachof the single enhancement lines in the double-array cavitysplits into two lines in the single-array cavity.

We now investigate this difference in more detail. Toavoid duplicity in the discussions, we will from now onfocus on TCD, which is typically simpler to measure thanACD. Figures 4(a,c) show frequency cuts out of the plotsin Figs. 3(a,b) at L=21.52 µm. The enhancements are nowshown on a linear rather than on a logarithmic scale. Forthe single-array cavity, Fig. 4(a) shows that all the modalenhancement lines are split, including the lowest frequencymode as seen in the inset, and that their splitting grows asthe frequency increases. The same is true for the double-array cavity in Fig. 4(c) except that the lowest frequencymode is not split. To explain these observations we considerthe frequency responses αω

TE and αωTM of a pair of TE

and TM resonances with the same amplitude and separatedby a frequency splitting ∆ω . We assume that they bothhave a Lorentzian lineshape with linewidth γc. The helicitypreservation (hω

c ) and helicity flip (hωf ) response of the

combination of resonances is

hωc =

αωTE +αω

TM√2

, hωf =

αωTE−αω

TM√2

, (3)

respectively. Figures 4(b,d) show the normalized helicitycontent (|hω

c |2 − |hωf |2) for different normalized separations

∆ω/(2πγc) as a function of the normalized angular frequencyω/ω0, where ω0 is the mean of the two resonance frequencies.The helicity content is normalized to the maximum valuetaken by |hω

c |2 − |hωf |2 in the case ∆ω = 0. In achiral

structures, the helicity content is a good proxy for the CD

signal29. For the larger ∆ω/(2πγc) values in Fig. 4(c), thehelicity is preserved, flipped, and then preserved again asthe frequency changes. This sequence can be understoodconsidering Eq. (3) and the change of π that the phase ofthe individual TE (TM) response undergoes when crossingits resonance. As ∆ω/(2πγc) → 0, the two resonancesprogressively degenerate into a single polarization preservingresonance. The ability of the model to reproduce the differentmode line structures in Figs. 4(a,b) strongly suggests thefollowing explanation. The TE/TM degeneracy in the modelat ∆ω/(2πγc) = 0 corresponds to the limit of reflection at 90degrees with respect to the surface normal inside the cavities.In both cavities, such degeneracy is progressively degraded bythe increase of the incidence angle of the first order modes asthe frequency increases for a given L (Eq. S10 in the Supp.Mat. of Ref. 30): The incidence angles inside the cavity forthe modes in Figs. 4(a,b) are approximately 85, 80, 75, and71 degrees (computed with Eq. (B9) in Ref. 30). The lastmode is not visible in Fig. 4(a). As the frequency decreases,the shapes in Figs. 4(a,c) can be reproduced by the modelusing an increasing ∆ω/(2πγc). In the case of the single-array cavity, small incidence angles are inherently present dueto the silicon slab and prevent ∆ω/(2πγc) to become smallenough to produce a single helicity preserving peak. Thelatter actually happens for the first mode of the double-arraycavity, as seen in the inset. Additionally, the coupling to theFabry-Perot modes is another factor that can contribute to thesplitting in the double-array cavity, for the reasons discussedabove.

A. Operational considerations

Once the structure of the enhancement lines has beenunderstood, let us now concentrate on a particular regionand consider some practical aspects of the operation of thecavities. Figure 5 shows cuts out of the plots in Fig. 3 inboth frequency and cavity length. The cuts are marked withred crosses in Figs. 3(a,b), which are centered at L=19.99 µmand f =35.97 THz. The enhancements are shown in linearunits. Figure 5 and Figs. 4(c,d) suggest that the single-array cavity could be operated in two different ways forsensing chiral molecules: Using one of the two positiveenhancement peaks, or using the negative enhancement regionin between them and inverting the resulting CD signal. Thelinewidths of lasers and the resolution of spectrometers74 inthis frequency region suggest that the operation using the areaof negative enhancement is straightforward, and that selectingthe positive peaks is more challenging because their widthis comparable to the resolution of standard spectrometers of≈0.015 THz. Such resolution is much finer than the widthof the enhancement line of the double-array cavity shownin Fig. 5(a). Regarding the cavity length L, the precisionneeded to select each peak is experimentally feasible usinglinear actuators with high precision. While the narrowerpeaks in Fig. 5(b) have a linewidth of about 0.15 µm, thereare commercially available linear actuators with a minimalincremental motion of 0.05 µm. Nevertheless, the simulations

8

FIG. 5: Plots of the data cuts through the red crosses in Figs. 3(a,b) comparing the TCD enhancement (in linear units here)of the double-array cavity design with the simplified single-array and silicon slab design. The cuts marked in the figure arecentered at L=19.99 µm and f =35.97 THz. This figure suggests that the single-array cavity can be operated in two ways forchiral molecule sensing: Using one of the two positive enhancement peaks or using the negative enhancement region in betweenthem, followed by inverting the resulting CD signal.

assume that the slab and the array are perfectly parallel toeach other. As discussed above, the degree of parallelismbetween the two sides of a cavity can be controlled by usinga set of three linear actuators in a triangular formation atthe edges of the cavity to adjust one of the sides (see theinset in Fig. 2). While the numerical simulation of a residualinclination between the two sides is challenging, its effectsare likely to be akin to those due to the cavity length varyingacross the transverse dimensions. The inclination shouldhence be carefully controlled to avoid averaging across thepositive and negative enhancements seen in Fig. 5(b) for thesingle-array cavity. The uniform sign of enhancement inthe double-array cavity makes it more robust against this.Finally, we note that both cavity designs can acquire somedegree of chirality depending on the kind of inclination.The end effect of this chirality needs to be experimentallyinvestigated. Overall, the considered operational requirementsare more stringent for the single-array cavity. This partiallycounteracts the benefits of such simplified design, which doesnot require the lateral alignment of two disk arrays with deepsub-micrometer precision.

Finally, we foresee the fabrication of silicon-disk arrayswith a lateral extent close to 20 mm. For a length L=20 µm,the cavity would hold 0.8 mL of analyte.

We will now address the realistic modeling of chiralmolecular resonances which is needed to accurately predictthe performance of our CD enhancing cavities.

III. CHIRAL MOLECULE RESONANCE MODEL WITHEXPERIMENTAL INPUT

Constitutive relations of the type written in Eq. (1)are routinely assumed for the numerical evaluation of CDenhancement systems17,25,27,29,30,34,35. To the best of our

knowledge, a frequency-dispersive model in accordance withenergy conservation [Eq. (3.219) in Ref. 75] and fullydetermined by available CD measurements has not yet beenreported. Such a model is needed to elucidate possible effectsdue to the interplay between the resonances of the moleculesand the resonances of the sensing system. Even when thedifference in the lifetimes of the two kinds of resonances islarge, suggesting a correspondingly small energy exchangerate between them, the use of realistic parameters is importantbecause effects that would otherwise be negligible can playa significant role in systems that achieve an unusually largelight-molecule interaction time, like our cavity designs. Inparticular, the settings that we used in Ref. 30 εω

eff = (2.14+i10−4)ε0, µω

eff = (1+ i10−4)µ0, and κωeff = i10−4, meet energy

conservation but also implicitly assume that the moleculeshave an unrealistic degree of duality symmetry, that is, thatthe handedness of the incident light changes much less duringthe light-molecule interaction than for real chiral molecules.A foreseeable effect of the helicity change is the progressivedegradation of the polarization purity of the light in themolecular solution. Intuitively, the CD signal decreases withdecreasing polarization purity and vanishes in the limit whenthe light is a perfect mix of both helicities. More surprisingly,we will show in the next section that the lack of dualitysymmetry of the molecules can significantly disturb the cavitymodes. None of these effects will be noticeable when theinteraction time between the light and the molecules is muchsmaller that the inverse of the rate of change of helicity inthe solution, like in typical CD spectrometers or non-cavitybased resonant CD enhancement systems. We now presentthe mentioned model.

We consider a homogeneous and isotropic solution ofidentical chiral molecules. Our objective is to describe thefrequency-dependent constitutive relations of such medium inspectral proximity to a resonance of the molecule. To such

9

end, we will consider the constitutive relations in Eq. (1) andfind expressions for the scalars εω

eff, µωeff, and κω

eff by meansof a microscopic model and its subsequent homogenization.The microscopic light-molecule interaction description thatwe consider is the molecular linear polarizability matrix[

pω(r)mω(r)

]=

[αω

ee αωem

αωme αω

mm

][Eω(r)Hω(r)

]= α

ω

[Eω(r)Hω(r)

], (4)

where the electromagnetic field oscillating at frequency ω

incident on a molecule at position r induces electric andmagnetic dipolar moments pω(r) and mω(r), respectively,according to a 6×6 complex matrix αω . This matrix fullydetermines the linear light-molecule interaction to dipolarorder, including both scattering and absorption of the incidentlight. In our case, the random orientation of the moleculesin the solution justifies the assumption of isotropy at themicroscopic level. This means that each of the αω

xy in Eq. (4)can be assumed to be the 3×3 identity matrix multiplied by acomplex scalar, which we denote by the same αω

xy symbols ina slight abuse of the notation that we adopt from now on.

We now assume the quasi-static model for the dipolarresonance of a chiral object from Eq. (20) in Ref. 76, which,after imposing isotropy reads

αω =

[αω

ee αωem

αωme αω

mm

]=

V1− ω2 + iγ ω

[ηee iηem−iηme ηmm

], (5)

where the ηxy are real, semi-positive, unitless, and frequency-independent, ω = ω

ω0, γ = γ

ω0, and ω0 and γ are the resonance

frequency and linewidth of the chiral molecule resonance,respectively. The factor V has units of volume in the naturalunits of Tab. I in Ref. 76. Additionally, we use the conditionin Eq. (6.126) of Ref. 77

ηeeηmm = η2em, (6)

which is met when the electric and magnetic dipolar momentsoriginate from the same current distribution. Then, afterdefining β = ηmm

ηee, we can write

αω =

V ηee

1− ω2− iγ ω

[1 is

√β

−is√

β β

], (7)

where s = {−1,+1} is a sign factor that determinesthe handedness of the resonance. Opposite molecularenantiomers will feature opposite signs. Note that, ourassumptions imply that β ≥ 0. When β = 0, the resonanceis purely electric and there is no chiral response. A changeto the helicity basis for both the electromagnetic fields andthe induced dipoles66 shows that when β = 1 the resonanceis excited only by electromagnetic fields of a fixed helicitydetermined by s, and that, upon interaction, the helicity of thescattered field is the same as that of the incident field. Whenβ = 1 the resonance is hence maximally electromagneticallychiral56 and dual symmetric66. Values of β > 1 correspond tochiral resonances whose magnetic component is larger thanthe electric component, and β → ∞ to a purely magneticresonance.

Equation (7) can be changed to SI units using Tab. I inRef. 76:

αω =

4πV ηee

1− ω2− iγ ω

ε is√

β

c

−is√

β

c β

, (8)

where ε = εrε0 is the electric permittivity of the solvent,which we assume to be non-dispersive in the vicinity of theresonance. The solvent is also assumed to be achiral and non-magnetic (κ = 0, µ = µ0). The speed of light in the solvent isc = 1/

√εµ .

At this point, the common homogenization equations fordilute mixtures of small chiral inclusions embedded in adielectric media [see e.g. Eq. (6.127) in Ref. 77] can be used78

to obtain the following expressions for the quantities in Eq. (1)

εωeff = ε

(1+

4π

3V

1− ω2− iγ ω

)µ

ωeff = µ

(1+

4π

3V β

1− ω2− iγ ω

),

κωeff =

4π

3V s√

β√

εr

1− ω2− iγ ω.

(9)

The unitless parameter V = ρV ηee collects all the factors thatdetermine the magnitude of the influence of the molecules:V and ηee from the microscopic resonance model, and themacroscopic concentration of molecules ρ . The parameter Vcan hence be varied for the study of a given CD enhancementsystem. We will now show how all the other parameters of themicroscopic model that appear in Eq. (9), namely s, β , and γ ,can be obtained from existing experimental measurements.

In typical CD spectrometers, a fixed volume of themolecular solution is subsequently illuminated by propagatinglight beams of opposite polarization handedness (±) coveringa certain frequency bandwidth. The absorption Aω

± in eachcase is measured as a function of the frequency. Their averageand difference are typically reported as a function of thefrequency, as well as the CD:

ΣAω =Aω++Aω

−2

, ∆Aω = Aω+−Aω

−, CDω =∆Aω

ΣAω. (10)

With this definition, the absolute value of the CDω0 is equal tothe Kuhn’s dissymmetry factor67. There is a large amount ofchiral spectroscopic measurements available in the literaturecovering many different molecules and proteins79–83. Let usassume that we are interested in a particular resonance of aparticular chiral molecule. The parameters ω0 and γ = γ/ω0in the model can immediately be determined from the centralfrequency and linewidth of the available CD measurementsof the molecular resonance. To show how s and β aredetermined, we take again a microscopic point of view andconsider a molecule at point r. Using Eq. (5) in Ref. 29 andEq. (8), the resonant absorption of the molecule at ω =ω0 canbe written as

Aω0± (r) = |Gω0

± (r)|2 2πω0εV ηee

γ

(1+β

2± s√

β

). (11)

10

where√

2Gω0± (r) = Eω0(r) ± iZHω0(r) are a version of

the Riemann-Silberstein vectors45,84. The experimentallymeasured absorbances Aω0

± are obtained after integratingEq. (11) on the appropriate volume, and accounting for themolecular concentration. Then, using Eqs. (10) and (11) it isstraightforward to show that

CDω0 =∆Aω0

ΣAω0 = s4√

β

1+β, (12)

which determines s and β

s = sign{CDω0} , β =

[2

|CDω0 |−

√4

|CDω0 |2−1

]2

. (13)

In this way, the resonance specific parameters ω0, γ , s,and β are all fixed by experimental data, and only V is leftfree. For our later purposes, it is convenient to relate V to themodulus of the constitutive chiral parameter at the resonancefrequency |κω0

eff |. From the last line of Eq. (9) we obtain that

V =3

4π

|κω0eff |γ√

β√

εrµr. (14)

IV. SENSING OF CHIRAL MOLECULES IN A CAVITY

We now use the model to predict the TCD enhancementand the TCD signals of the different cavities for a particularresonance of binol. We use the VCD measurementsfrom a commercial spectrometer (Fig. 4 in Ref. 69) toextract the following parameters: ω0/(2π) = f0 =44.21 THz,γ =0.258 THz, CDω0 = 1.34×10−3, s =+1, and obtain β =1.12× 10−7. We also adapt the following parameters of thecavity to the target frequency [see Figs. 1(b,c)]: h =1.082 µm,r =1.574 µm, a =4.721 µm, and a relative permittivity equalto 1.8912 for the substrate and the solvent. We use the lowestfrequency mode in both cavities and a cavity length equal toL =11.52 µm(12.80 µm) for the single(double) array cavity.We consider four different values of |κω0

eff | ∈ [10−8,10−7,7.5×10−7,10−6], which imply four different concentrations ρ

proportional to each value of |κω0eff | according to Eqs. (14,9).

Besides binol (yellow lines in Figs. 6,7 with CDω0 = 1.34×10−3), we also analyze the results for CDω0 = 1.34× 10−4

(purple lines) and CDω0 = 2 (red lines), which allow usto study the effects of different degrees of duality breaking(helicity change) of the molecules. It can be deduced fromEq. (13) that the perfect duality condition β = 1 is achievedwhen CDω0 = ±2, and that the departure from dualityincreases as |CDω0 | decreases.

Figures 6 and 7 show the TCD enhancement and theTCD signal for the double-array and single-array cavity,respectively. Let us first analyze the results for the double-array cavity. We start with the perfectly dual CDω0 = 2case. The plots in Figs. 6(a)-(d) show that the enhancement isindependent of the concentration. In particular, the maximumenhancement of ≈ 310 is always achieved at the molecular

resonance frequency. Additionally, the plots of the absolutevalue of the TCD signal in Figs. 6(e)-(h) are identical to eachother except for the expected scaling with |κω0

eff |. This showsthat the effect of the cavity is independent of the molecularconcentration if there is no helicity change in the solution.For CDω0 = 2 the cavity is always working in an undisturbedfashion. In contrast, the TCD enhancement and the spectralsignature of the TCD signal depend on |κω0

eff | for the non-dualcases (CDω0 6= 2). The previously discussed reduction of theCD signal due to the light changing helicity upon interactionwith the molecules can at most bring the enhancement to zeroand does not explain the sign changes that we observe inFigs. 6(b-d). These can nonetheless be explained by noticingthat the shapes of the enhancement lines in Figs. 6(b-d) areessentially the same as those in Figs. 4(c,d), indicating that thehelicity change due to the molecules causes a split of the TEand TM resonances of the cavity. We note that the undisturbedcavity resonance for the double-array cavity is always helicitypreserving, as can be deduced from the results with lowest|κω0

eff | in Fig. 6(a). The plots in Fig. 6 suggest that the splittingis a function of ρ/CDω0 . For example, the shape changesthat can be seen for CDω0 = 1.34× 10−3 when going from|κω0

eff | = 10−7 to |κω0eff | = 10−6, are identical to the changes

that can be seen for CDω0 = 1.34× 10−4 when going from|κω0

eff |= 10−8 to |κω0eff |= 10−7.

A similar analysis holds for the corresponding results forthe single-array cavity in Fig. 7, except that the undisturbedsingle cavity resonance is already split, as shown by theCDω0 = 2 lines. Then, there is an additional split whichincreases with ρ/CDω0 . Another difference is that themaximum enhancement (≈ 2100 for the CDω0 = 2 case), isnever achieved in the case of binol, whose largest positiveenhancement is ≈ 850 for the lowest value of |κω0

eff |. On theother hand, the region of negative enhancement, featuring anenhancement of ≈ −35 at its center, is very similar for binoland for the perfectly dual case when |κω0

eff | ≤ 10−7. Theseobservations are consistent with ρ/CDω0 having a larger(smaller) disturbing effect for thiner (wider) enhancementlinewidths, i.e. larger (smaller) light-matter interaction times.

Let us now further discuss the absolute value of theTCD measurements shown in Figs. 6,7(e-h) for the case ofbinol. Whether the actual TCD value is above or below themeasurement detection threshold determines the feasibilityof the sensing experiment. For discussion purposes, weassume a sensitivity threshold of 10−5, which is close tothe 8×10−6 noise level of some commercially available CDspectrometers3,74. The threshold is drawn as a green-dashedline in the bottom rows of Figs. 6,7. For |κω0

eff | ≤ 10−7,the reference signal without the cavities (blue line) is wellbelow the detection threshold. In these cases, both cavitiesare capable of bringing the TCD signal above the threshold.

We note that the chiral resonance model can be easilyextended to solutions containing both enantiomers of themolecule. It suffices to modify the values of the concentrationρ and of the CD

ρ → ρL +ρD, CDω0 → ρL−ρD

ρL +ρD|CDω0 |, (15)

11

FIG. 6: Results for the double-array cavity. Panels (a)-(d) show the TCD enhancement for different values of |κω0eff | (see

titles). In each of the panels, the TCD enhancement for different values of CDω0 are shown (see legend). Panels (e)-(h) show theabsolute value of the TCD for the reference case without the arrays (blue lines) and for the cavity (red, yellow, and purple lines).The green-dashed line marks a detection threshold (see text).

where ρL(ρD) is the concentration of the L(D) mirror versionof the chiral molecule and ρL−ρD

ρL+ρDis commonly known as the

enantiomeric excess (ee).Finally, we note that the CD enhancing cavities can also

work as optical rotation enhancing systems after appropriatechanges in the illumination and measurements. Preliminarysimulation results show that the OR enhancements are similarto the discussed CD enhancements, which are smaller thanthe four orders of magnitude OR enhancements predictedfor the multi-pass CRDP technique14. Also, at theircurrent sizes, CRDP setups offer an advantageous frequency-independent enhancement because their free-space cavitylengths are longer than the coherence length of light. Thisprevents the frequency-dependent constructive and destructiveinterference that leads to the typical mode build-up in shortercavities.

V. CONCLUDING REMARKS

We have numerically demonstrated that different opticalcavities can resonantly enhance the vibrational circulardichroism (VCD) signal of solutions of chiral molecules

by one to three orders of magnitude, for a given moleculeconcentration and given thickness of the cell containingthe molecules. In particular, we have compared a newlyintroduced simplified cavity based on a single silicon diskarray and a silicon film with a previously introduced designfeaturing two arrays of silicon disks. In this article, we haveused a realistic model for the electromagnetic response of thesolution of chiral molecules at the frequency region containinga molecular resonance. This refined model has allowed us touncover a rather surprising effect: The structure of the cavitymodes can be affected by the lack of duality symmetry of themolecules, that is, by the partial change of the handedness oflight upon light-molecule interaction. For sufficiently largeratios of the molecular concentration divided by the Kuhn’sdissymmetry factor of the molecular resonance, the dualitybreaking of the molecules causes a split of the TE and TMcomponents of the cavity modes. This split is in additionto other TE/TM splits that are inherent to the design of thecavities, and disturb the ideal degeneracy featured by perfectlyhelicity-preserving modes.

The experimental implementation of our CD enhancementcavities should extend the capabilities of traditionalchiral spectrometers to unprecedentedly small volumesof analyte. In this respect, substituting the sample cell

12

FIG. 7: Results for the single-array cavity. Panels (a)-(d) show the TCD enhancement for different values of |κω0eff | (see titles).

For improved visibility, the frequency range on the horizontal axes of the top panels is a zoomed-in portion of the frequencyrange in the bottom panels. In each of the panels, the TCD enhancement for different values of CDω0 are shown (see legend).Panels (e)-(h) show the absolute value of the TCD for the reference case without the arrays (blue lines) and for the cavity (red,yellow, and purple lines). The green-dashed line marks a detection threshold (see text).

inside CD spectrometers by our cavities is a straightforwardimplementation path which is currently possible, and that canbe used as a stepping stone towards applications on futurelab-on-a-chip devices.

ACKNOWLEDGMENTS

This research has been funded by the Hector FellowAcademy, by the Deutsche Forschungsgemeinschaft (DFG,German Research Foundation) under Germany’s ExcellenceStrategy via the Excellence Cluster 3D Matter Made to Order(EXC-2082/1 – 390761711) and via the SFB 1173 (Project-ID258734477), by the Carl Zeiss Foundation, by the HelmholtzAssociation via the Helmholtz program “Materials SystemsEngineering” (MSE), and by the KIT through the “VirtualMaterials Design” (VIRTMAT) project. X.G.-S. is pursuinghis Ph.D. within the Karlsruhe School of Optics and Photonics(KSOP) and acknowledges financial support. Finally, we aregrateful to the company JCMwave for their free provision ofthe FEM Maxwell solver JCMsuite.

DATA AVAILABILITY

The data that supports the findings of this study areavailable within the article.

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